Postgres on conflict do update on composite primary keys

Just place both keys in the ON CONFLICT clause:

INSERT INTO answer VALUES (1,1,'q1') 
ON CONFLICT (person_id,question_id) 
DO UPDATE SET answer = EXCLUDED.answer; 

Example:

INSERT INTO answer VALUES (1,1,'q1') 
ON CONFLICT (person_id,question_id) 
DO UPDATE SET answer = EXCLUDED.answer;             

SELECT * FROM answer;
 person_id | question_id | answer 
-----------+-------------+--------
         1 |           1 | q1
(1 Zeile)

INSERT INTO answer VALUES (1,1,'q1-UPDATED') 
ON CONFLICT (person_id,question_id) 
DO UPDATE SET answer = EXCLUDED.answer;             

SELECT * FROM answer;
 person_id | question_id |   answer   
-----------+-------------+------------
         1 |           1 | q1-UPDATED
(1 Zeile)

Demo: db<>fiddle

You could also define the primary externally to the table and then you don't need to re-write all columns included therein.

CREATE TABLE "answer" (
  "person_id" integer NOT NULL REFERENCES person(id), 
  "question_id" integer NOT NULL REFERENCES question(id) ON DELETE CASCADE, /* INDEXED */
  "answer" character varying (1200) NULL);

ALTER TABLE "answer" ADD CONSTRAINT answer_pk PRIMARY KEY (person_id, question_id);

And then:

INSERT INTO answer VALUES (1,1,'q1') ON CONFLICT ON CONSTRAINT answer_pk DO UPDATE SET answer = EXCLUDED.answer;

When the constraint changes in the future you don't need to manually adjust the insert statements to reflect this.